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<h1>Computing a sparse solution of a set of linear inequalities</h1>
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<pre class="codeinput">
<span class="comment">% Section 6.2, Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% "Just relax: Convex programming methods for subset selection</span>
<span class="comment">%  and sparse approximation" by J. A. Tropp</span>
<span class="comment">% Written for CVX by Almir Mutapcic - 02/28/06</span>
<span class="comment">%</span>
<span class="comment">% We consider a set of linear inequalities A*x &lt;= b which are</span>
<span class="comment">% feasible. We apply two heuristics to find a sparse point x that</span>
<span class="comment">% satisfies these inequalities.</span>
<span class="comment">%</span>
<span class="comment">% The (standard) l1-norm heuristic for finding a sparse solution is:</span>
<span class="comment">%</span>
<span class="comment">%   minimize   ||x||_1</span>
<span class="comment">%       s.t.   Ax &lt;= b</span>
<span class="comment">%</span>
<span class="comment">% The log-based heuristic is an iterative method for finding</span>
<span class="comment">% a sparse solution, by finding a local optimal point for the problem:</span>
<span class="comment">%</span>
<span class="comment">%   minimize   sum(log( delta + |x_i| ))</span>
<span class="comment">%       s.t.   Ax &lt;= b</span>
<span class="comment">%</span>
<span class="comment">% where delta is a small threshold value (determines what is close to zero).</span>
<span class="comment">% We cannot solve this problem since it is a minimization of a concave</span>
<span class="comment">% function and thus it is not a convex problem. However, we can apply</span>
<span class="comment">% a heuristic in which we linearize the objective, solve, and re-iterate.</span>
<span class="comment">% This becomes a weighted l1-norm heuristic:</span>
<span class="comment">%</span>
<span class="comment">%   minimize sum( W_i * |x_i| )</span>
<span class="comment">%       s.t. Ax &lt;= b</span>
<span class="comment">%</span>
<span class="comment">% which in each iteration re-adjusts the weights W_i based on the rule:</span>
<span class="comment">%</span>
<span class="comment">%   W_i = 1/(delta + |x_i|), where delta is a small threshold value</span>
<span class="comment">%</span>
<span class="comment">% This algorithm is described in papers:</span>
<span class="comment">% "An Affine Scaling Methodology for Best Basis Selection"</span>
<span class="comment">%  by B. D. Rao and K. Kreutz-Delgado</span>
<span class="comment">% "Portfolio optimization with linear and &iuml;&not;?xed transaction costs"</span>
<span class="comment">%  by M. S. Lobo, M. Fazel, and S. Boyd</span>

<span class="comment">% fix random number generator so we can repeat the experiment</span>
seed = 0;
randn(<span class="string">'state'</span>,seed);
rand(<span class="string">'state'</span>,seed);

<span class="comment">% the threshold value below which we consider an element to be zero</span>
delta = 1e-8;

<span class="comment">% problem dimensions (m inequalities in n-dimensional space)</span>
m = 100;
n = 50;

<span class="comment">% construct a feasible set of inequalities</span>
<span class="comment">% (this system is feasible for the x0 point)</span>
A  = randn(m,n);
x0 = randn(n,1);
b  = A*x0 + rand(m,1);

<span class="comment">% l1-norm heuristic for finding a sparse solution</span>
fprintf(1, <span class="string">'Finding a sparse feasible point using l1-norm heuristic ...'</span>)
cvx_begin
  variable <span class="string">x_l1(n)</span>
  minimize( norm( x_l1, 1 ) )
  subject <span class="string">to</span>
    A*x_l1 &lt;= b;
cvx_end

<span class="comment">% number of nonzero elements in the solution (its cardinality or diversity)</span>
nnz = length(find( abs(x_l1) &gt; delta ));
fprintf(1,[<span class="string">'\nFound a feasible x in R^%d that has %d nonzeros '</span> <span class="keyword">...</span>
           <span class="string">'using the l1-norm heuristic.\n'</span>],n,nnz);

<span class="comment">% iterative log heuristic</span>
NUM_RUNS = 15;
nnzs = [];
W = ones(n,1); <span class="comment">% initial weights</span>

disp([char(10) <span class="string">'Log-based heuristic:'</span>]);
<span class="keyword">for</span> k = 1:NUM_RUNS
  cvx_begin <span class="string">quiet</span>
    variable <span class="string">x_log(n)</span>
    minimize( sum( W.*abs(x_log) ) )
    subject <span class="string">to</span>
      A*x_log &lt;= b;
  cvx_end

  <span class="comment">% display new number of nonzeros in the solution vector</span>
  nnz = length(find( abs(x_log) &gt; delta ));
  nnzs = [nnzs nnz];
  fprintf(1,<span class="string">'   found a solution with %d nonzeros...\n'</span>, nnz);

  <span class="comment">% adjust the weights and re-iterate</span>
  W = 1./(delta + abs(x_log));
<span class="keyword">end</span>

<span class="comment">% number of nonzero elements in the solution (its cardinality or diversity)</span>
nnz = length(find( abs(x_log) &gt; delta ));
fprintf(1,[<span class="string">'\nFound a feasible x in R^%d that has %d nonzeros '</span> <span class="keyword">...</span>
           <span class="string">'using the log heuristic.\n'</span>],n,nnz);

<span class="comment">% plot number of nonzeros versus iteration</span>
plot(1:NUM_RUNS, nnzs, [1 NUM_RUNS],[nnzs(1) nnzs(1)],<span class="string">'--'</span>);
axis([1 NUM_RUNS 0 n])
xlabel(<span class="string">'iteration'</span>), ylabel(<span class="string">'number of nonzeros (cardinality)'</span>);
legend(<span class="string">'log heuristic'</span>,<span class="string">'l1-norm heuristic'</span>,<span class="string">'Location'</span>,<span class="string">'SouthEast'</span>)
</pre>
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<pre class="codeoutput">
Finding a sparse feasible point using l1-norm heuristic ... 
Calling sedumi: 200 variables, 100 equality constraints
   For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 100, order n = 201, dim = 201, blocks = 1
nnz(A) = 5200 + 0, nnz(ADA) = 2650, nnz(L) = 1375
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            8.50E-01 0.000
  1 :   3.98E+01 3.79E-01 0.000 0.4456 0.9000 0.9000   6.27  1  1  6.6E+00
  2 :  -5.85E+00 1.47E-01 0.000 0.3882 0.9000 0.9000   1.78  1  1  1.9E+00
  3 :  -2.47E+01 6.78E-02 0.000 0.4609 0.9000 0.9000   1.49  1  1  7.5E-01
  4 :  -3.26E+01 2.55E-02 0.000 0.3769 0.9000 0.9000   1.32  1  1  2.5E-01
  5 :  -3.46E+01 1.01E-02 0.000 0.3936 0.9000 0.9000   1.13  1  1  9.5E-02
  6 :  -3.55E+01 3.53E-03 0.000 0.3513 0.9000 0.9000   1.05  1  1  3.3E-02
  7 :  -3.58E+01 1.30E-03 0.000 0.3695 0.9000 0.9000   1.02  1  1  1.2E-02
  8 :  -3.59E+01 2.58E-05 0.000 0.0198 0.9258 0.9000   1.01  1  1  1.9E-03
  9 :  -3.59E+01 6.19E-06 0.000 0.2397 0.9000 0.0000   1.00  1  1  8.5E-04
 10 :  -3.59E+01 1.39E-06 0.000 0.2251 0.9000 0.8569   1.00  1  1  2.0E-04
 11 :  -3.59E+01 1.50E-07 0.000 0.1077 0.9138 0.9000   1.00  1  1  2.9E-05
 12 :  -3.59E+01 2.89E-08 0.000 0.1924 0.9180 0.9000   1.00  1  1  6.1E-06
 13 :  -3.59E+01 3.29E-09 0.000 0.1139 0.9000 0.9112   1.00  1  1  6.8E-07
 14 :  -3.59E+01 4.33E-12 0.000 0.0013 0.9990 0.9990   1.00  1  1  
iter seconds digits       c*x               b*y
 14      0.1   Inf -3.5940776942e+01 -3.5940776942e+01
|Ax-b| =   2.7e-15, [Ay-c]_+ =   8.9E-15, |x|=  7.1e+00, |y|=  1.0e+01

Detailed timing (sec)
   Pre          IPM          Post
1.000E-02    6.000E-02    0.000E+00    
Max-norms: ||b||=1, ||c|| = 2.494989e+01,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.58293.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +35.9408

Found a feasible x in R^50 that has 44 nonzeros using the l1-norm heuristic.

Log-based heuristic:
   found a solution with 44 nonzeros...
   found a solution with 44 nonzeros...
   found a solution with 43 nonzeros...
   found a solution with 43 nonzeros...
   found a solution with 41 nonzeros...
   found a solution with 40 nonzeros...
   found a solution with 39 nonzeros...
   found a solution with 39 nonzeros...
   found a solution with 37 nonzeros...
   found a solution with 37 nonzeros...
   found a solution with 37 nonzeros...
   found a solution with 37 nonzeros...
   found a solution with 37 nonzeros...
   found a solution with 37 nonzeros...
   found a solution with 37 nonzeros...

Found a feasible x in R^50 that has 37 nonzeros using the log heuristic.
</pre>
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